Arithmetic function

In number theory, an arithmetic function is a function defined on the set of positive integers, usually with integer, real or complex values.

Classes of arithmetic function

Arithmetic functions which have some connexion with the additive or multiplicative structure of the integers are of particular interest in number theory.

Multiplicative functions

We define a function a(n) on positive integers to be

• Totally multiplicative if $a(mn)=a(m)a(n)$ for all m and n.
• Multiplicative if $a(mn)=a(m)a(n)$ whenever m and n are coprime.

The Dirichlet convolution of two arithmetic function a(n) and b(n) is defined as

$a\star b(n)=\sum _{d\mid n}a(d)b(n/d).\,$

If a and b are multiplicative, so is their convolution.

Examples

• Carmichael's lambda function
• Euler's totient function
• Jordan's totient function
• Möbius function

See also

• Formal Dirichlet series
• Average order of an arithmetic function
• Normal order of an arithmetic function